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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">H</font>, <font color="Olive" title="b4">K</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b4">K</font> or for <font color="Olive" title="b5">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b4">K</font> or  not <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b4">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">K</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b2">K</font> or for <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b2">K</font> or  not <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b2">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">H</font>, <font color="Maroon" title="c5">K</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )</span><br/>









<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1"/>
( for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) or  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_2"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_2"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">h1</font>, <font color="Maroon" title="c8">k1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E6:16_1_2"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font>
 <span class="kw">and </span><br/><a NAME="E7:16_1_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c7">h1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E8:16_1_2"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c8">k1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">h2</font>, <font color="Maroon" title="c10">k2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E10:16_1_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font>
 <span class="kw">and </span><br/><a NAME="E11:16_1_2"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c9">h2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E12:16_1_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_2"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>;<br/><a NAME="E13:16_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">h</font>, <font color="Maroon" title="c13">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_2_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_2_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1_2_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>;<br/>
<a NAME="E6:16_1_2_1"/>
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b1">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Olive" title="b2">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_2_1_1"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>;<br/>
<a NAME="E2:16_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>;<br/>
<a NAME="E3:16_1_2_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E21">A13</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:16_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/>
<span class="kw">take </span><font color="Maroon" title="c14">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Olive" title="b1">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c15">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c15">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<a NAME="E5:16_1_2_1_1"/>
<font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E6:16_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_2_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<a NAME="E7:16_1_2_1_1"/>
<font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E8:16_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16_1_2_1_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c15">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E3:16_1_2_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E5:16_1_2_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E4:16_1_2_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E6:16_1_2_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:16_1_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E15:16_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_2_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">h</font>, <font color="Maroon" title="c13">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_2_2"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_2_2"/><span class="lab"><font color="Green" title="E19">A17</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1_2_2"/><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>;<br/>
<a NAME="E6:16_1_2_2"/>
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b1">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Olive" title="b2">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1_2_2_1"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>;<br/>
<a NAME="E2:16_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>;<br/>
<a NAME="E3:16_1_2_2_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E22">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2_2_1"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:16_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">k1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/>
<span class="kw">take </span><font color="Maroon" title="c14">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Olive" title="b1">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c15">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">k1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c15">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )</span><br/>

<a NAME="E5:16_1_2_2_1"/>
<font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E6:16_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A21</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_2_2"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<a NAME="E7:16_1_2_2_1"/>
<font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E8:16_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16_1_2_2_1"/>
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c15">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:16_1_2_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E5:16_1_2_2"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="txt" href="#E4:16_1_2_2_1"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="txt" href="#E6:16_1_2_2_1"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:16_1_2_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_8.html#T17" target="_self" title="GROUP_8:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:16_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E17:16_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> or  not <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c5">K</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_1_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
